Hydrodynamical loop models have been traditionally used as a tool to understand the physics and the thermodynamics of coronal loops on the sun and the stars, in particular during complex and energetic phenomena such as flares. There have been many applications of hydrodynamical loop models to the study of confined solar plasma (Nagai 1980; Peres et al. 1982; Cheng et al. 1983; Doscheck et al. 1983; Cheng et al. 1984; Nagai & Emslie 1984; Fisher et al. 1985a-c; MacNeice 1986; Mariska 1987). The Palermo-Harvard code (henceforth, PH, Peres et al. 1982), in particular, has been extensively applied to many problems of coronal physics, including studies of loop stability, analysis of physical processes influencing the thermal flare (Pallavicini et al. 1983; Peres & Reale 1993a,b; Reale & Peres 1995), diagnostics of the location of the energy release in solar flares (Peres et al. 1987; Antonucci et al. 1987; Antonucci et al. 1993); the code has also been used to model a stellar flare observed by the Einstein satellite (Reale et al. 1988) allowing to infer the characteristic length of the flaring structure, to study the decay phase of solar (Serio et al. 1991; Jakimiec et al. 1992; Sylwester et al. 1993) and stellar flares (Reale et al. 1993), in order to devise diagnostics of flare conditions from the decay phase, and to study microflares (Peres et al. 1993).
The lack of resolution in the chromosphere-corona transition region is
a problem shared by almost all existing time-dependent hydrodynamical
loop models, including the PH code. In the transition region the
temperature rises by two orders of magnitude to its typical coronal
values (
K), over a distance range shorter than 100 km,
which is a fraction 
% or less of typical loop lengths; the
density undergoes a symmetrical decrease by two orders of magnitude.
These very steep temperature and density gradients complicate the
numerical resolution of the equations. Furthermore the transition
region is strongly variable and dynamic: for instance it rapidly moves
and steepens during flares and transient phenomena observable in the
corona.
Since appropriate diagnostics of the very steep transition region requires the study of many optically thin spectral lines emitted mostly in the EUV band, which originate there, well sampled hydrodynamic solutions are indispensable for a correct interpretation of the data from space instruments such as those on board SOHO. As we will show, a good resolution in the transition region may also influence the solution in the corona; in fact, the evaluation of the conductive flux and mass motions in the transition region also determines the quality of the results in the rest of the atmosphere.
Here we present an updated version of the PH code, which describes the transition region with a resolution adequate to the requirements. To achieve this goal we have devised a regridding algorithm with non-uniform spacing which samples and accurately computes temperature, density and velocity everywhere in the loop system and maintains this resolution during the evolution by changing the grid with time so as to adapt it to the evolving solution. A uniform grid with high spatial resolution would not be appropriate, because of the inordinately high number of points and computer resources needed. A fixed grid with an unequal distribution of points would equally be inappropriate since the transition region rapidly moves and steepens during a typical flare evolution: after a few seconds of time integration an initially well distributed grid would no longer remain adequate.
Some authors have used a re-adaptive grid to compute the equations during the evolution of a flare but limited their analysis to the first few seconds, and in fact at best only to the rising phase of the event. In a few cases these limitations were necessary because a complete computation required an unaffordable amount of computer time (McClymont et al. 1983; Fisher et al. 1985; MacNeice 1986), in others the simulations were subject to immediate numerical instability (Gan et al. 1991).
In the previous version of the Palermo-Harvard numerical code (Peres et al.
1982), the hydrodynamical equations, translated into difference equations,
are solved on a fixed spatial grid whose spacing changes along the spatial
domain with a logarithmic law, finer in the chromosphere and coarser
in the corona. Such a grid assures a number of points in the transition
region larger than that obtainable with a uniform grid with an equal total
number of points thus providing a partial solution to the need of spatial
resolution. During the first phases of a flare, in fact, the steepened
transition region typically moves towards the lower atmosphere where the
spacing is finer. This choice of grid, however, is only partially
satisfactory: temperature and density gradients are usually so steep
during a flare that in practice, when using 256 points along a typical
active region coronal loop of half-length 
, only two or
three grid points are in the transition region.
In order to satisfy these requirements we have devised an algorithm which generates a self-adaptive grid so as to maintain a fractional variation of temperature, density and pressure between adjacent grid points always below a chosen value. We will show how the implementation of this adaptive grid has significantly improved the results of the PH code. In Sect. 2 we describe the regridding algorithm, showing sample simulations in Sect. 3 and examining the synthesis of spectra in Sect. 4; in Sect. 5 we discuss our results and draw our conclusions.